Magnetic Resonance in Quantum Information.ppt · • 1975-1990 Advanced multiAdvanced multi-pulse...
Transcript of Magnetic Resonance in Quantum Information.ppt · • 1975-1990 Advanced multiAdvanced multi-pulse...
Magnetic Resonance in Quantum InformationInformation
Christian DegenSpin Physics and Imaging groupSpin Physics and Imaging group
Laboratory for Solid State Physics
www.spin.ethz.ch
ContentContent
F t f ( l ) tiFeatures of (nuclear) magnetic resonanceBrief History of NMRy
E l 1Example 1:Liquid-state NMR (=> Factoring of 15)
Example 2:Example 2:Quantum information using single spins in diamond
Magnetic nucleiMagnetic nucleiBlue => Nuclei with spin I = ½R d N l i i h i I ½Red => Nuclei with spin I > ½Grey => Nuclei with spin zero, or not known
Time and Energy scales in NMR & EPRTime and Energy scales in NMR & EPR
El tN l i
R l t Z 43 MH /T 28 GH /T
ElectronsNuclei(Protons)
• Relevant energy: Zeeman energy 43 MHz/T 28 GHz/Tin magnetic field (up to ~24Tesla) (~0.2µeV/T) (~120µeV/T)
• Polarization (room temperature) ~10-5 ~1%• Spin relaxation time T1 0.1…105 s 10-9…1 sSpin relaxation time T1 0.1…10 s 10 …1 s• Spin coherence time T2 ~1…100µs (solids) 10-9…10-3 s
~10ms…10s (liquids)• Spin rotations (pulses) ~1-10µs ~1-100 ns• Swaps (spin-spin couplings) ~10µs…10ms ~0.1-10µs
NMR largely satisfies the DiVincenzo criteriaQ bit l i ½ i B fi ld (↑ d ↓ 0 d 1)Qubits: nuclear spins ½ in B0 field (↑ and ↓ as 0 and 1)
Quantum gates: RF pulses and delay times
( ) Input: Boltzmann distribution (room temperature)
Readout: detect spin states with RF coilReadout: detect spin states with RF coil
Coherence times: easily several seconds
(Gershenfeld & Chuang 1997,Cory Havel & Fahmi 1997)Cory, Havel & Fahmi 1997)
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Brief history of NMRBrief history of NMRLiterature suggestion:
A AbragamA. Abragam,“Time reversal: An autobiography”(Oxford University Press, 1989)
• 1922 Stern-Gerlach experiment• 1939 Rabi’s molecular beam experiment [Nobel Prize 1944]• 1945-1950 Bloch equations [Nobel Prize 1952]• 1950-1965 Physical basis of NMR: Relaxation, Couplings, DNP, etc.• ~1965 Fourier-transform NMR (Ernst) [Nobel Prize 1991]
1973 Invention of magnetic resonance imaging [Nobel Prize 2003]• 1973 Invention of magnetic resonance imaging [Nobel Prize 2003]• ~1975 Multi-dimensional NMR • 1975-1990 Advanced multi-pulse techniques (decoupling )1975 1990 Advanced multi pulse techniques (decoupling, …)• 1980- Application in Structural Biology [Nobel Prize 2002]• 1995 - Application in Quantum engineeringpp g g
E l 1 Li id t t NMRExample 1: Liquid-state NMR
Principles of NMR QC
Techniques for qubit control
State of the art & Outlook
=> “NMR Quantum Computing”
Slides courtesy of Lieven VandersypenThen: IBM Almaden, Stanford University
Now: Kavli Institute of NanoScience TU DelftNow: Kavli Institute of NanoScience, TU Delft
Liquid state NMRLiquid-state NMR
Survey of NMR quantum computing
Principles of NMR QC
T h i f bit t lTechniques for qubit control
Summary: Pros & ConsSummary: Pros & Cons
Nuclear spin Hamiltonian withoutNuclear spin HamiltonianMultiple spins qubit/qubit
coupling
MHz1H 500 ~ 25 mK
chemical shiftsof the five F qubits
1H 500 ~ 25 mK13C 126
N 5115N -5119F 47031P 202(at 11.7 Tesla)bit l l tiqubit level separation
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Hamiltonian with RF fieldHamiltonian with RF fieldsingle-qubit rotations
σz σx σyσz σx σy
rotating wave approximation typical strength Ix, Iy : up to 100 kHz
Rotating frame Lab frame 7
Nuclear spin HamiltonianNuclear spin HamiltonianCoupled spins J>0: antiferro mag.
J<0: ferro magcoupling term J<0: ferro-mag.
Typical values: J up to few 100 Hz
16 configurations
lid (d h d) li ( ) l d l lsolid (dashed) lines are (un)coupled levels8
M ki t t i l k ldMaking room temperature spins look coldwarm ↑↑ ↑↓ ↓↑ ↓↓
(Cool to mK)Optical pumpingDNP
Effective pure state(Gershenfeld&Chuang, Science ‘97,Cory Havel & Fahmi PNAS ‘97)DNP, …
↑↑
Cory, Havel & Fahmi, PNAS 97)
equalize population↑↑↑↑
↑↓ ↓↑ ↓↓ quasi coldcold ↑↓ ↓↑ ↓↓cold ↑↓ ↓↑ ↓↓
Look exactly like cold spins !
warmwarmquasi cold
[Hz] [Hz] 10
Effective pure state preparationEffective pure state preparation(1) Add up 2N-1 experiments (Knill,Chuang,Laflamme, PRA 1998)( ) p p ( , g, , )
↑↑ ↑↓ ↓↑ ↓↓ ↑↑ ↑↓ ↓↑ ↓↓ ↑↑ ↑↓ ↓↑ ↓↓ ↑↑ ↑↓ ↓↑ ↓↓
+ + =Later : (2N - 1) / N experiments (Vandersypen et al., PRL 2000)
prepare equal population (on average) and look atdeviations from equilibrium.
(2) Work in subspace (Gershenfeld&Chuang, Science 1997)
↑↑↑ ↑↑↓ ↑↓↑ ↑↓↓ ↓↑↑ ↓↑↓ ↓↓↑ ↓↓↓ ↑↑↑ ↑↑↓ ↑↓↑ ↑↓↓ ↓↑↑ ↓↑↓ ↓↓↑ ↓↓↓
compute with qubit states that have the same energyand thus the same population. 11
Read out in NMRRead-out in NMR
Phase sensitive detection
|0�|0� yx x
positive signal for |0> (in phase)positive signal for |0> (in phase)z
|1�
yY90 z
|1�x
y yx
negative signal for |1> (out of phase) 12
NMR SpectrometerNMR Spectrometer
Transmission /Transmission / Receiver coil
Computer – Console – Superconducting magnetComputer Console Superconducting magnet
Liquid state NMRLiquid-state NMR
Survey of NMR quantum computing
Principles of NMR QC
T h i f bit t lTechniques for qubit control
Summary: Pros & ConsSummary: Pros & Cons
Off-resonance pulses and spin-selectivity
spectral content of a square pulse
off-resonant pulses induce eff.σ rotation in addition to σσz rotation in addition to σx,y
may induce transitions in otherqubits 17
Pulse shaping for improvedspin-selectivityp y
gaussian pulse profile gaussian spectrum
l t lkless cross-talk
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Mi i li SMissing coupling terms: Swap
How to couple distant qubits with only nearest neighborphysical couplings?
Missing couplings: swap states along qubit network
SWAP CNOT CNOT CNOTSWAP12 = CNOT12 CNOT21 CNOT12
1 2 3 4
“only” a linear overhead ...
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Undesired couplings: refocusUndesired couplings: refocusremove effect of coupling during delay times
opt. 1: act on qubit B
XB180 XB180
opt. 2: act on qubit A• There exist efficient extensions for arbitrary coupling networks• Refocusing can also be used to remove unwanted Zeeman terms 20
C i lComposite pulsesE l Y X YExample: Y90X180Y90
corrects for corrects forunder/over-rotation
zoff-resonance
z
x yx
x yy
However: doesn’t work for arbitrary input stateBut: there exist composite pulses that work for all input statesBut: there exist composite pulses that work for all input states
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M l l l iMolecule selection
A quantum computer is a known molecule.It d i d tiIts desired properties are:
spins 1/2 (1H, 13C, 19F, 15N, ...)long T1’s and T2’sheteronuclear, or large chemical shifts
d J li t k ( l k d)good J-coupling network (clock-speed)
stable available soluble required to makestable, available, soluble, ...spins of sametype addressable
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Liquid state NMRLiquid-state NMR
Survey of NMR quantum computing
Principles of NMR QC
T h i f bit t lTechniques for qubit control
Summary: Pros & ConsSummary: Pros & Cons
The good newsThe good news
Quantum computations have been demonstrated in the labQuantum computations have been demonstrated in the lab
A high degree of control was reached, permitting hundreds ofoperations in sequence
A variety of tools were developed for accurate unitary controlover multiple coupled qubits
useful in other quantum computer realizations
Spins are natural, attractive qubits
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The main issue: ScalingThe main issue: Scaling
We do not know how to scale liquid NMR QC
Main obstacles:Main obstacles:• Signal after initialization ~ 1 / 2n [at least in practice]
C• Coherence time typically goes down with molecule size• We have not yet reached the accuracy threshold ...
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Main sources of errors in NMR QC
Early on (heteronuclear molecules)inhomogeneity RF field
L t (h l l l )Later (homonuclear molecules)J coupling during RF pulses
Finallydecoherencedecoherence
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Electron spin qubits
SL SRSL SR
Kane Nature 1998 Loss & DiVincenzo PRA 1998Kane, Nature 1998 Loss & DiVincenzo, PRA 1998
Gruber, Science, 1997
Quantum information using single spins in diamonddiamond
Literature hint:F. Jelezko, J. Wrachtrup,, p,
”Single defect centres in diamond: A review”Phys. stat. sol. (a) 203, 3207 (2006).
Single spins in diamondSingle spins in diamond
Diamond nitrogen vacancy center (NV Center)- Principles of Optically-detected magnetic resonance (ODMR)
I iti li ti i l ti d t f NV l t i- Initialization, manipulation, read-out of NV electron spin
Environment: Diamond host materialEnvironment: Diamond host material
Applicationspp
Optically detected magnetic resonance (ODMR)Optically-detected magnetic resonance (ODMR)
Id D t t ti l i t d f i / di f h tIdea: Detect optical instead of microwave/radio-frequency photons
Requirement: Optical photon must be correlated with spin stateRequirement: Optical photon must be correlated with spin state
σ = ↑or ↓
Intensity I Intensity I’(σ)Intensity IWavelength λPolarization P
Intensity I (σ)Wavelength λ’(σ)Polarization P’(σ)
Optically detected magnetic resonanceOptically detected magnetic resonanceSome History
• Optical detection of paramagnetism in phenantren (Kwiram, 1967)
(… 1980ies: Invention of single molecule spectroscopy …)
• Optical detection of magnetic resonance in a single pentacene moleculeWrachrup, Moerner (1993)
• Optical detection of single nitrogen vacancy centers in diamond:G b W ht (1997)Gruber, Wrachtrup (1997)
Single spin detection of NV centersSingle spin detection of NV centers
G b S i 1997Gruber, Science 1997
Readout and Polarization of the NV centerReadout and Polarization of the NV centerConduction band
5.5 eV0
1±3E
Readout:• ms = 0 scatters 30% more light than m = ±1
1A0
630‐800 nm
532 nm
1 945 V
than ms = ±1• Many readout cycles needed to detect spin state
>100 ns
630 800 nm
10…20 ns
1.945 eV
Polarization (“Reset”):• Illumination pumps spin into m =
2 9 GHz
0
1±3A
Illumination pumps spin into ms = 0 state (>90% fidelity)
2.9 GHz
Valence band
Spin HamiltonianSpin Hamiltonian
Zero field splitting(2.87 GHz)
Electron Zeeman(2.8 MHz/Gauss)
Hyperfine coupling to 14N (~2.2 MHz)(2.87 GHz) (2.8 MHz/Gauss) ( 2.2 MHz)
Jelezko, Journal of Phys. Cond Mat. 2006
NV centers in diamondNV centers in diamondSome properties:p p
• Single photon emitter
Optically stable:• Optically stable:No bleaching, no blinking
High quantum yield 70%
Nitrogen Vacancy defect in diamond
• High quantum yield ~70%
• Room temperatureNitrogen Vacancy defect in diamond
• Long spin coherence times >1ms
• In nanodiamonds down to 5 nm
• Excellent chemical stability
Non to ic biomarkers• Non-toxic biomarkers
• Inexpensive!
• …
Fabrication Ion implantationFabrication – Ion implantationNV centers implanted into single crystal diamond, then patterned with waveguide
Toyli, Nano Letters, 2010
Rapid spin manipulation by waveguidesRapid spin manipulation by waveguidesRabi Oscillations
Spin inversion within ~ 1 ns
Fuchs, Science, 2009
Single spins in diamondSingle spins in diamond
Diamond nitrogen vacancy center (NV Center)- Principles of Optically-detected magnetic resonance (ODMR)
I iti li ti i l ti d t f NV l t i- Initialization, manipulation, read-out of NV electron spin
Environment: Diamond host materialEnvironment: Diamond host material
Applicationspp
Spin environmentSpin environment
N l i El t i
• Nearby N defect electron spins • 14NV nuclear spin (I=1)
Nuclear spins Electron spins
(S=1/2) (“P1 or C centers”)• Nearby other NV center (S=1)
• 13C spins (I=½) 1.03% natural abundance
Fi t h ll 3 d li b d • …• Surface spins (electronic, nuclear)
– First shell: 3x dangling bonds– Second, third, … shell
Distant “Classical” spin bath– Distant: “Classical” spin bath• …• Surface spins (adsorbates)• Surface spins (adsorbates)
Other electronic spinsOther electronic spinsNearby N defect with S = ½ (“P1 center”)
Gaebel, Nature Physics, 2006
The 13C nuclear spin environmentThe 13C nuclear spin environment
Near 13C spins – resolved linesNear C spins – resolved lines
Intermediate 13C spins – lines not resolved, but still moderate hyperfine coupling
Distant 13C spins – very weak hyperfine coupling (“classical ensemble”)
Specific nearby 13CSpecific nearby 13C
Two qubits: NV + 13C Three qubits: NV + 13C + 13CTwo qubits: NV + C Three qubits: NV + C + C
Dutt, Science, 2007
Repetitive readoutRepetitive readoutStore and Readout NV spin state in nearby 13C nucleus
calculation store readout ninit Example:calculation store readoutinit Example:Rabi oscillation
Jiang, Science, 2009
Distant 13C spin bathDistant 13C spin bathPeriodic “revivals” of spin echo with 13C Larmor frequency
Childress, Science, 2006
Eliminating 13C nuclear spin bathEliminating 13C nuclear spin bathThe materials scientists’ way …
13C content Spin coherence time T2
20.7% 0.010 ms8.4 % 0.12 ms1.1 % (natural) 0.65 ms0.4% 1.8 ms
Balasubramanian, Nature Materials, 2009
Eliminating 13C nuclear spin bathEliminating 13C nuclear spin bath… the spectroscopists way
Multi‐pulse decoupling ( l “d i l d li ”)(also “dynamical decoupling”)
Carr‐Purcell‐Meiboum‐Gibbs sequence
Single spins in diamondSingle spins in diamond
Diamond nitrogen vacancy center (NV Center)- Principles of Optically-detected magnetic resonance (ODMR)
I iti li ti i l ti d t f NV l t i- Initialization, manipulation, read-out of NV electron spin
Environment: Diamond host materialEnvironment: Diamond host material
Applicationspp
Hybrid implementationsHybrid implementationsNV centers to superconducting microwave resonator
Arrangement Transmission vs. Frequency
Kubo, PRL, 2010
Application in nanoscale magnetic imagingApplication in nanoscale magnetic imaging
Measure nanoscale magnetic fields from:Measure nanoscale magnetic fields from:• Single electron spins
• Nuclear spins in (bio)moleculesNuclear spins in (bio)molecules• Superconductors
• Magnetic nanostructuresMagnetic nanostructures• Quantized currents in mesoscopics
Degen, Appl. Phys. Lett, 2008 Disk drive (Rondin, arXiv, 2011)